# Forbidden paths and cycles in ordered graphs and matrices

János Pach, G. Tardos

Research output: Contribution to journalArticle

28 Citations (Scopus)

### Abstract

At most how many edges can an ordered graph of n vertices have if it does not contain a fixed forbidden ordered subgraph H? It is not hard to give an asymptotically tight answer to this question, unless H is a bipartite graph in which every vertex belonging to the first part precedes all vertices belonging to the second. In this case, the question can be reformulated as an extremal problem for zero-one matrices avoiding a certain pattern (submatrix) P. We disprove a general conjecture of Füredi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases when P is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the fact that the number of times that the unit distance can occur among n points in the plane is O(n 4/3).

Original language English 359-380 22 Israel Journal of Mathematics 155 https://doi.org/10.1007/BF02773960 Published - 2006

### Fingerprint

Cycle
Path
Disprove
Extremal Problems
Graph in graph theory
Bipartite Graph
Subgraph
Verify
Unit
Alternatives
Zero
Vertex of a graph

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

In: Israel Journal of Mathematics, Vol. 155, 2006, p. 359-380.

Research output: Contribution to journalArticle

title = "Forbidden paths and cycles in ordered graphs and matrices",
abstract = "At most how many edges can an ordered graph of n vertices have if it does not contain a fixed forbidden ordered subgraph H? It is not hard to give an asymptotically tight answer to this question, unless H is a bipartite graph in which every vertex belonging to the first part precedes all vertices belonging to the second. In this case, the question can be reformulated as an extremal problem for zero-one matrices avoiding a certain pattern (submatrix) P. We disprove a general conjecture of F{\"u}redi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases when P is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the fact that the number of times that the unit distance can occur among n points in the plane is O(n 4/3).",
author = "J{\'a}nos Pach and G. Tardos",
year = "2006",
doi = "10.1007/BF02773960",
language = "English",
volume = "155",
pages = "359--380",
journal = "Israel Journal of Mathematics",
issn = "0021-2172",
publisher = "Springer New York",

}

TY - JOUR

T1 - Forbidden paths and cycles in ordered graphs and matrices

AU - Pach, János

AU - Tardos, G.

PY - 2006

Y1 - 2006

N2 - At most how many edges can an ordered graph of n vertices have if it does not contain a fixed forbidden ordered subgraph H? It is not hard to give an asymptotically tight answer to this question, unless H is a bipartite graph in which every vertex belonging to the first part precedes all vertices belonging to the second. In this case, the question can be reformulated as an extremal problem for zero-one matrices avoiding a certain pattern (submatrix) P. We disprove a general conjecture of Füredi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases when P is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the fact that the number of times that the unit distance can occur among n points in the plane is O(n 4/3).

AB - At most how many edges can an ordered graph of n vertices have if it does not contain a fixed forbidden ordered subgraph H? It is not hard to give an asymptotically tight answer to this question, unless H is a bipartite graph in which every vertex belonging to the first part precedes all vertices belonging to the second. In this case, the question can be reformulated as an extremal problem for zero-one matrices avoiding a certain pattern (submatrix) P. We disprove a general conjecture of Füredi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases when P is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the fact that the number of times that the unit distance can occur among n points in the plane is O(n 4/3).

UR - http://www.scopus.com/inward/record.url?scp=33845632978&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845632978&partnerID=8YFLogxK

U2 - 10.1007/BF02773960

DO - 10.1007/BF02773960

M3 - Article

AN - SCOPUS:33845632978

VL - 155

SP - 359

EP - 380

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -